Automotive powertrains require accurate control systems to meet government regulations and customer expectations. Manufacturers of powertrain control systems recognize the benefit of applying analytical control system design methods to the development and calibration of their products. Accurate control systems provide improved performance, reduced variation, and reduced calibration effort.
Analytical control system design applies Control Theory to vehicle systems using differential equation engineering (or difference equation engineering, for discrete time systems). Generally, analytical control system design includes modeling a physical system as a differential equation and analytically determining a new differential equation. The new differential equation is often called the “control law” or “control algorithm.” The control law is connected to the system in question to obtain a desired behavior. The desired behavior is often expressed as either a differential equation or some attribute of a differential equation.
The analytic theory of linear differential and difference equations and the design theory for linear control systems are particularly well-suited for systems that can be described by linear, constant coefficient equations. The theory of non-linear differential equations, however, is less helpful for analytic design of control systems. Generally, the theory of non-linear differential equations is largely confined to describing equation properties that can ensure existence, uniqueness, and stability of solutions. As a result, the analytic design theory for non-linear control systems usually involves problems in terms of stability.
“Scheduling” involves using linear system descriptions for non-linear systems by modeling a nonlinear system as a linear system with parameters that are functions of other variables. In other words, linear systems are scheduled to represent non-linear behavior. Scheduling is extremely powerful because it permits the application of linear analysis and design techniques to non-linear systems. Conventional methods for scheduling linear systems include table look-up (i.e. interpolation) scheduling, polynomial scheduling, and “jump” linear scheduling.
In table look-up scheduling, system parameters are interpolated between values stored in a one-dimensional or multi-dimensional table. Scheduling via table lookup can be computationally expensive and does not permit standard least squares identification of table entries. Moreover, the computational expense increases geometrically with the dimension of the scheduling space (i.e. increases with additional scheduling variables). The inability to use least squares identification obscures the relationship between model parameters (as stored in the table entries) and system behavior. This limits the ability of designers to apply extensions of standard linear design theory, such as least squares identification.
In polynomial scheduling, system parameters are polynomial functions of the scheduling variables. In “jump” linear scheduling, system parameters are held constant over a range of the scheduling variables, but can change discretely when crossing into a different range of the scheduling parameters. “Jump” linear scheduling is computationally efficient, and it preserves transparency between system parameters and performance. For example, “jump” linear scheduling permits the use of least squares identification. However, with “jump” linear scheduling, the system behavior can become discontinuous across scheduling boundaries. Such discontinuity can produce undesirable system performance.